Hund-Mulliken method

Van Vleck(i67) has compared from the quantitative viewpoint the advantages and disadvantages of the Slater-Pauling method and the Hund-Mulliken method for methane, CH4. He also shows, by applying both methods, that the regular tetra-... 

Like Hund, Mulliken developed the basic Schrodinger equation in the direction of establishing the electron charge density resulting from a combination of the attractions of two or more nuclei and the averaged repulsions of other electrons in the system. This is a method that favors some particular region of space and disfavors others. In contrast to the Heitler-London method, it over-emphasizes, rather than underemphasizes, the ionic character of a molecule. For example, for the H2 molecule, Hund s wave function equation assumes that it is just as probable to have two electrons around the same nucleus as to have one electron around each nucleus. For a molecule made up of identical nuclei, this treatment is a considerable exaggeration of the ionic character of the molecule. 

Hiickel s application of this approach to the aromatic compounds gave new confidence to those physicists and chemists following up on the Hund-Mulliken analysis. It was regarded by many people as the simplest of the quantum mechanical valence-bond methods based on the Schrodinger equation. 66 Hiickel s was part of a series of applications of the method of linear combination of atom wave functions (atomic orbitals), a method that Felix Bloch had extended from H2+ to metals in 1928 and that Fowler s student, Lennard-Jones, had further developed for diatomic molecules in 1929. Now Hiickel extended the method to polyatomic molecules.67... 

Slater developed an approach (the "determinantal method") that offers a way of choosing among linear combinations (essentially sums and differences) of the polar and nonpolar terms in the Hund-Mulliken equations to bring their method into better harmony with the nonpolar emphasis characteristic of the Heitler-London-Pauling approach in which polar terms do not figure in the wave equation. 72... 

Dans le probleme de la construction d une molecule, selon la methode de Hund-Mulliken-Hiickel on ecrit, en approximation d ordre zero les auto-fonctions moleculaires comme des combinaisons lineaires des auto-fonctions electroniques des atomes constituants. De toutes les combinaisons possibles, celles qui con-viennent au probleme doivent etre conformes aux representations irreductibles du groupe de symetrie de la molecule. [71]... 

Huckel s second method also known as the HMH method (Hund-Mulliken-Huckel) fits on to the treatment of the electrons in an atom and in a metal (Chapter IV). Each electron is assumed to be moving in a field due to the nuclei and the other electrons. The total wave function of the electrons is then a product of these one-electron functions. 

The problems for quantum chemists in the mid-forties were how to improve the methods of describing the electronic structure of molecules, valence theory, properties of the low excited states of small molecules, particularly aromatic hydrocarbons, and the theory of reactions. It seemed that the physics needed was by then all to hand. Quantum mechanics had been applied by Heitler, London, Slater and Pauling, and by Hund, Mulliken and Hiickei and others to the electronic structure of molecules, and there was a good basis in statistical mechanics. Although quantum electrodynamics had not yet been developed in a form convenient for treating the interaction of radiation with slow moving electrons in molecules, there were semi-classical methods that were adequate in many cases. 

Even if no integral parametrizations are introduced and the HF equations are all correctly solved, the method eventually turns out to be theoretically incomplete. Despite the correct treatment of electronic exchange (X) within Hartree-Fock theory, electronic correlation (C) is totally missing. This is easily shown for the case of the H2 molecule in which we use the bonding solution of the H2 molecular ion ( + = cr from Equation (2.15)) to build up an antisymmetrized molecular wave function. This means that we put both electrons (ri and rz) of the H2 molecule into the same ip+ orbital, and Pauli s principle is obeyed by means of the ct/ spinors. Neglecting orbital overlap and any pre-factors, for simplicity, the so-called Hund-Mulliken [124] (another name... 

Most quantitative molecular calculations are done using the MO method, because it is computationally much simpler than the VB method. The MO method was developed by Hund, Mulliken, and Lennard-Jones in the late 1920s. Originally, it was used largely for qualitative descriptions of molecules, but the electronic digital computer has made possible the calculation of accurate MO functions (Section 13.17). 

Wheland, G. W. /. Chem. Phys. 1934,2,474 argued that the valence bond method as described by Heitler, London, Slater, and Pauling gives results closer to experimentally observed values than does the molecular orbital method attributed to Hund, Mulliken, and Hiickel. 

The method of indicating the electronic states deviates a little from the one which has been usual up to now, which was introduced by Hund and Mulliken. It appears more efficient for the treatment of the electronic structure of two-atomic molecules with a larger number of electrons. While the Hund-Mulliken designation for the electronic states in a two-atomic molecule shows in which states they... 

In 1931 Hiickel published the first in a series of four papers in which he applied the new quantum mechanics to the benzene problem using for the first time the VB method for aromatic compounds. Little reference is now made to this part of Hiickel s paper. In the second part of this same paper, he applied to aromatics a method that Bloch had used for crystal lattices. In this method electrons are placed into orbitals that extend over the entire molecule instead of being localized on a single atom. These molecular orbitals may in turn be constructed as linear combinations of atomic orbitals, and such an approach is usually called the Hund—Mulliken—Huckel (HMH) or molecular orbital (MO) method. Huckel s work in this second part of the paper leads to the famous 4/7 -f 2 rule for monocyclic conjugated hydrocarbons, though we do not find it explicitly stated here. 

The other approach, proposed slightly later by Hund[9] and further developed by Mulliken[10] is usually called the molecular orbital (MO) method. Basically, it views a molecule, particularly a diatomic molecule, in terms of its united atom limit . That is, H2 is a He atom (not a real one with neutrons in the nucleus) in which the two positive charges are moved from coinciding to the correct distance for the molecule. HF could be viewed as a Ne atom with one proton moved from the nucleus out to the molecular distance, etc. As in the VB case, further adjustments and corrections may be applied to improve accuracy. Although the imited atom limit is not often mentioned in work today, its heritage exists in that MOs are universally... 

In a molecule, the one-electron eigenfunctions (Mulliken and Hund s molecular orbitals, M. 0.) are determined by a core field U(x,y,z) and have well-defined symmetry type yn. However, the actual calculation of such M. 0. is very difficult in polyatomic molecules, whereas the Hartree-Fock method can be applied to monatomic entities when large electronic computers are available (13,14). I have written a book about several of the principal problems regarding the concept of one-electron functions... 

All quantum chemical calculations are based on the self-consistent field (SCF) method of Hatree and Fock (1928-1930) and the MO theory of Hund, Lennard-Jones, and Mulliken (1927-1929). A method of obtaining SCF orbitals for closed shell systems was developed independently by Roothaan and Hall in 1951. In solving the so-called Roothan equations, ab initio calculations, in contrast to semiempirical treatments, do not use experimental data other than the values of the fundamental physical constants. 

Molecular orbital theory originated from the theoretical work of German physicist Friederich Hund (1896-1997) and its apphcation to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. MuUiken (1896-1986) (Hund, 1926, 1927a, b Mulliken, 1926, 1928a, b, 1932). Inspired by the success of Heitler and London s approach, Finklestein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894-1954) later suggested that only valence electrons need be treated as delocalized inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones, 1929). 

Mulliken and Hund, along with important contributions from Herzberg and Lennard-Jones, developed an alternative method that does not give the localized electron pair special... 

I think the problem was that nobody understood the two-electron bond until Heitler and London s research in 1926. Then Hund and Mulliken developed the molecular orbital method. It s true that Lewis lived much beyond 1926, but Lewis himself did not understand the nature of the electron-pair bond in his early research. Failing that, I think his time sort of passed. It may have been a good idea to give him the prize but I can understand why they didn t, because Lewis himself certainly didn t formulate the physical basis of the bond. 

Molecular-orbital theory, especially Hiickel theory (1931) and its extensions, is by now widely appreciated by chemists in all areas of research and teaching. This was not always so. Actually, the method of linear combinations of atomic orbitals was used by Bloch in 1928 for wave functions in crystals. Molecular wave-functions were introduced by Mulliken (1928), Hund (1928), Herzberg (1929) and Lennard-Jones (1929) for diatomic molecules, and the extensions to polyatomic molecules followed quickly thereafter. 

In 1927, Burrau calculated the energy of Hj and Heitler and London treated the hydrogen molecule. In 1928, the Heitler-London or valence bond method was applied to many electron systems, and simultaneously Hund and Mulliken started the development of the molecular orbital theory. In 1931, Slater expressed the v/avefunctions of complex molecules in terms of Slater determinants made up of linear combinations of atomic orbitals. Thus, the Golden Age was born. 

In subsequent independent papers, Pauling [4] and Slater [6] generalized the valence-bond treatment made for the H2 molecule to polyatomic systems as H2O, NH3, CH4 etc. .. where an atom of the first period (the second row) is linked to hydrogens by several two-electron bonds they described the valence orbitals coming from the central atom by appropriate s and p combinations known later as hybrid orbitals. At the same time Hund [7] and Mulliken [8] presented another quantum theory of valence, the molecular orbital method in LCAO form, using the spectroscopic concept of molecular configuration built from s, p, d. ..pure atomic orbitals. The actual status of the hybridization process was clarified by Van Vleck [9], who showed that the various approximations... 

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